1. Field of the Invention
This invention relates to space situational awareness and more particularly to a passive and autonomous system and method for simultaneously determining the position and velocity % of both tracking and target space-based vehicles from line-of-sight measurements.
2. Description of the Related Art
Space vehicle navigation done for satellites and/or probes uses a number of technologies such as optical trackers, radars, GPS etc. Though all these methods work, most of them require the active participation of ground based Earth assets to complete a solution. As the number of satellites in orbit and other space vehicles on non-orbiting trajectories continues to proliferate exponentially, the demands upon these aforementioned ground based assets are increasing in kind. To reduce the demands for expensive infrastructure upgrades, more satellites are commonly executing some form of autonomous navigation.
GPS is a commonly used autonomous navigation technique, but it becomes increasingly difficult to implement beyond the radius of its 12 hour orbit. Larger antennas are required to receive signals from the far side of the Earth and/or local satellite backlobes, thus increasing system mass and cost. Although considered autonomous. GPS techniques do rely on the network of GPS satellites. A passive and truly autonomous solution which does not rely on any other assets or require the addition of costly and heavy hardware is welcome, especially for deep space satellite implementations.
Extended Kalman Filters (EKFs) are snidely implemented recursive filters used to infer satellite position and velocity states from incomplete measurements such as angle only data from optical trackers, Doppler information from RF transmissions or range (with poor angle information) data from ground based radars. See “An Introduction to the Kalman Filter” SIGGRAPH 2001, Los Angeles, Calif., August 12-17, pp. 1-46, 2001 for a detailed presentation on Kalman Filters and Extended Kalman Filters.
The generic Extended Kalman Filter (EKF) is widely used and documented in many places. The classic predictor/corrector equations can lead to near optimal results as long as the time step is small enough so that the system can be viewed as linear for the duration of the chosen time step.
Predict{circumflex over (x)}k|k-1=f({circumflex over (x)}k-1|k-1,uk,0)Pk|k-1=FkPk-1|k-1FkT+Qk Update{tilde over (y)}k=zk−h({circumflex over (x)}k|k-1,0)Sk=HkPk|k-1HkT+Rk Kk=Pk|k-1HkTSk−1 {circumflex over (x)}k|k={circumflex over (x)}k|k-1+Kk{tilde over (y)}k Pk|k=(I−KkHk)Pk|k-1 where {circumflex over (x)}k|k is the estimate of the state vector at time k, Pk|k is the error covariance matrix, uk is the control vector, Fk is the state transition matrix, Qk is the process noise covariance, zk is an observation of the true state xk, Hk is the measurement matrix, Rk is the measurement noise covariance, yk is the innovation, Sk is the innovation covariance, and Kk is the Kalman gain.The state transition matrix (Fk) and measurement (Hk) matrices are defined to be the following Jacobians
                                                        F              k                        =                                          ∂                f                                            ∂                x                                                                                                  x              ^                                                      k                -                1                            ❘                              k                -                1                                              ,                      u            k                              ⁢                          ⁢                        H          k                =                              ∂            h                                ∂            x                                              x      .              k      ❘              k        -        1            
FIGS. 1a through 1c demonstrate the use of an EKF for tracking a target satellite 10 using angle-only information from a notional optical tracker 12 located on the surface of the Earth 14. With known values for the position (Rtrk) and velocity (Vtrk) of the fixed optical tracker in an inertial frame of reference centered on the Earth, the EKF 16 infers target position (Rtgt) and velocity (Vtgt) from state vector xk over time using line-of-sight (LOS) measurements 17 and an Earth gravity model that dictates how the EKF expects the target to move within the effects of the Earth's gravitational field. Essentially the gravity model provides an acceleration at any distance to the center of the gravitational body. In this case, a simple single point gravitational model is used, though more complicated models can be used to improve propagation accuracy.
The EKF state vector x has nine filter states for the target; position, velocity and acceleration for each of the X, Y and Z axes. The coordinate system is defined as Earth Centered Inertial (ECI). The updated values of the three position states infer target position (Rtgt) and the three velocity states infer target velocity (Vtgt). The updated values of the three acceleration states infer target acceleration (Atgt) although that is not typically an output of interest.                x=[r(1) target x position estimate,                    r(2) target y position estimate,            r(3) target z position estimate,            v(1) target x velocity estimate.            v(2) target y velocity estimate,            v(3) target z velocity estimate,            a(1) target x acceleration estimate.            a(2) target y acceleration estimate.            a(3)]; target z acceleration estimatewhere r, v and a are the three dimensional position, velocity and acceleration vectors of the target vehicle. Initialization of these parameters, along with their covariance matrices, is done within relatively large bounds, with knowledge of the likely distance derived from brightness and/or other reasonable orbital limitations. Given that the tracker's position is known, the estimated relative position of the target to the tracker (rd) and its magnitude (rdmag) can be used to establish the H, or measurement matrix:H(1,1)=(rdmag2−rd(1)^2)/rdmag3.H(1,2)=−(rd(2)*rd(1))/rdmag3;H(1,3)=−(rd(3)*rd(1))/rdmag3;H(2,1)=−(rd(1)*rd(2))/rdmag3;H(2,2)=(rdmag2−rd(2)^2)/rdmag3;H(2,3)=−(rd(3)*rd(2))/rdmag3;H(3,1)=−(rd(1)*rd(3))/rdmag3;H(3,2)=−(rd(2)*rd(3))/rdmag3;H(3,3)=(rdmag2−rd(3)^2)/rdmag3;                        
Measurement noise, defined as the RMS vertical and horizontal accuracy of the LOS measurement (track_err), is established in line of sight coordinates, and then rotated into ECI coordinates for use as the “R” matrix specified by the EKF equations.
  Rlos  =      [                            1                          0                          0                                      0                                      track_err            ^            2                                    0                                      0                          0                                      track_err            ^            2                                ]  The line of sight noise Rlos is rotated into inertial coordinates.psi=tan−1(rd(2),rd(1));theta=tan−1(rd(3),norm(rd(1:2)));phi=0; andTilos=cosine—ypr(psi,theta,phi).where cosine_ypr produces a cosine matrix using the yaw, pitch and roll Euler angles corresponding to psi, theta and phi respectively. This gives R=Tilos*Rlos*Tilos′.
The state transition matrix F incorporates the effects of the perceived gravitational model in a 9×9 array. F is the Jacobian of f, the function which propagates x old to x new. Rex, y & z are the estimated position of the target relative to the gravitational body, which are needed to establish acceleration so that the target states can be propagated forward one time step (dt).Rex=x(1);Rev=x(2):Rez=x(3); andRe—mag2=Rex*Rex+Rev*Rev+Rez*Rez. 
      F    =          [                                    1                                0                                0                                dt                                0                                0                                              0.5              *                              dt                ^                2                                                          0                                0                                                0                                1                                0                                0                                dt                                0                                0                                              0.5              *                              dt                ^                2                                                          0                                                0                                0                                1                                0                                0                                dt                                0                                0                                              0.5              *                              dt                ^                2                                                                          0                                0                                0                                1                                0                                0                                dt                                0                                0                                                0                                0                                0                                0                                1                                0                                0                                dt                                0                                                0                                0                                0                                0                                0                                1                                0                                0                                dt                              ]        ;F(7:9,1:9)=0;F(7,1)=−G*m_earth*(Re—mag2^(− 3/2)−3*Rex^2*(Re—mag2^(− 5/2)));F(7,2)=3*G*m_earth*Rex*Rey*Re—mag2^(− 5/2):F(7,3)=3*G*m_earth*Rex*Rez*Re—mag2^(− 5/2);F(8,1)=3*G*m_earth*Rex*Rey*Re—mag2^(− 5/2);F(8,2)=−G*m_earth*(Re—mag2^(− 3/2)−3*Rey^2*(Re—mag2^(− 5/2)));F(8,3)=3*G*m_earth*Rey*Rez*Re—mag2^(− 5/2);F(9,1)=3*G*m_earth*Rex*Rez*Re—mag2^(− 5/2);F(9,2)=3*G*m_earth*Rex*Rez*Re—mag2^(− 5/2);F(9,3)=G*m_earth*(Re—mag2^(− 3/2)−3*Rez^2*(Re—mag2^(− 5/2)));where G is the gravitational constant and m_earth is the earth mass.
The case illustrated shows how target position and velocity errors 18 and 20, respectively, steadily decrease overtime as sequential line of sight measurements are fed into the filter. To make this technique work, the position and velocity of the optical tracker must be known (updated and accurate) relative to the gravitational center of the Earth so that estimated accelerations can be calculated correctly for the target. The existence of a gravitational well is crucial to the proper convergence of the filter because without it, there is no observable difference between targets passing by slowly at short range versus targets passing by quickly from long range. Filters of this type have been in common use for several decades and are an innate part of various space situational awareness programs.
FIGS. 2a through 2c extend the concept from a fixed ground based sensor to a space-based sensor 22 such as on a satellite in orbit about the Earth. Here the problem is effectively identical to the case shown in FIGS. 1a-1c except that tracker position (Rtrk) and velocity (Vtrk) are substantially more dynamic. Position is typically established with GPS, permitting accurate estimates of the tracker position and velocity over time so that the basic construct of the EKF is unchanged from the case in FIGS. 1a-1c. The tracker position and velocity are provided as inputs to the EKF. Although performance is not substantially different, an orbital tracker has a different angle history relative to the target, producing a slightly different convergence history 24 and 26. This approach does have the drawback of requiring additional capability (e.g. GPS antenna and processing) to estimate the tracker position and velocity, which is then fed into the EKF with the LOS measurements.